Method for designing phasor algorithms of phasor measurement units based on complex bandpass digital filters

ABSTRACT

A general design method for phasor estimation algorithms on different applications is described based on a complex finite impulse response (FIR) band-pass filter. To facilitate the design of the complex band-pass filter for different requirements and reduce the trial and error process, a design framework based on the error mathematical models is described. Using an absolute value inequality theorem, the general error models between the filter gain and the error limitations of all the valuables measured by phasor measurement units (PMUs) are established separately. The filter design criteria obtained by the error models can determine the passband and stopband gain range of complex band-pass filters.

BACKGROUND

Phasor measurement units (PMUs) can realize synchronized dynamicmonitoring of power systems, and thus, they are widely deployed intransmission and distribution networks. With the rapid development ofrenewables, flexible transmissions and active loads, power systems arebecoming increasingly complex. As a result, PMUs for different scenariosand applications are needed.

SUMMARY

It is an object of the present disclosure to provide for phasorestimation methods for different scenarios and applications. In oneexample embodiment, a general design method for phasor estimationalgorithms on different applications is described based on a complexfinite impulse response (FIR) band-pass filter. To facilitate the designof the complex band-pass filter for different requirements and reducethe trial and error process, a design framework based on the errormathematical models is described. Using an absolute value inequalitytheorem, the general error models between the filter gain and the errorlimitations of all the valuables measured by phasor measurement units(PMUs) are established separately. The filter design criteria obtainedby the error models can determine the passband and stopband gain rangeof complex band-pass filters.

Additionally, three phasor algorithms for different classes of PMUs areanalyzed, and their performance are compared with experimental tests.The results demonstrate that the described method can design phasoralgorithms to provide accurate measurements under different scenarios.In particular, when compared with the existing filter design criteria,the described error models have the following advantages. First, theamplitude error (AE) and phase error (PE) limitations are utilized inanalyzing the filter gain range. Second, the magnitude response of thedifferentiator is considered in the analysis of the frequency error (FE)and rate-of-change-of-frequency (ROCOF) error (RFE) limitations. Third,the suppression of the negative fundamental component is considered todetermine the stopband gain range. Fourth, the influence of frequencydeviation and reporting rate is considered for phase modulation signals,and thus the error models can be applied for both frequency-tracking andfixed filters. The test results show that the described method candesign different classes of phasor algorithms with excellent measurementperformance.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to facilitate a fuller understanding of the present invention,reference is now made to the attached drawings. The drawings should notbe construed as limiting the present invention, but are intended only toillustrate different aspects and embodiments of the invention.

FIG. 1 shows an example framework for a design method for phasorestimation based on complex band-pass filters.

FIG. 2 shows an example diagram of phasor estimation filters designed byIRWLS algorithm.

FIG. 3 shows example filtering characteristics and parameter ranges forM-class, P-class, and S-class filters.

FIG. 4 displays the phasor errors of an example S-class PMU in the AMand PM tests.

FIG. 5 shows example phasor errors obtained from interference tests.

FIG. 6 shows an example PMU prototype according to an exampleembodiment.

FIG. 7 illustrates exemplary hardware components for a server.

DETAILED DESCRIPTION

Exemplary embodiments of the invention will now be described in order toillustrate various features of the invention. The embodiments describedherein are not intended to be limiting as to the scope of the invention,but rather are intended to provide examples of the components, use, andoperation of the invention.

Phasor measurement units (PMUs) can realize synchronized dynamicmonitoring of power systems, and thus they are widely deployed intransmission and distribution networks. PMUs are classified into P- andM-classes in IEEE standards (“IEEE Std.”). The P-class PMU is suitablefor protection applications and needs fast response speed. Additionally,it has a narrow measurement band and no need for rejecting out-of-band(OOB) interferences. The M-class PMU has a wider measurement bandwidthand needs to suppress OOB interferences with different frequency bandsat different reporting rates.

However, with the rapid development of renewables, flexibletransmissions and active loads, power systems are becoming increasinglycomplex. As a result, PMUs for different scenarios and applications areneeded. For instance, there has been many reports of sub-synchronousoscillations (SSOs) in the areas with high penetration of renewables.PMUs have become one of the tools used to monitor SSOs in real timebecause of their high reporting rate and synchronization. Therefore, toconduct monitoring of power systems under different scenarios, andachieve their effective control, phasor estimation methods with goodperformance must be developed and studied for different classes of PMUs.

Some phasor estimation methods can be mainly classified into time-domainand frequency-domain methods. The time-domain methods, such as theGauss-Newton method and Kalman filter method, require accurate signalmodels. However, some signal models only contain fundamental andharmonic components without inter-harmonics, thereby resulting in a weakability to suppress out-of-band (OOB) inter-harmonics. Besides, somemethods use iterative techniques to calculate phasors, thereby causing aheavy computational burden. Furthermore, some time-domain methods maysuffer from the problem of numerical instability in engineeringapplications.

The frequency-domain methods may be used because of the simplicity oftheir calculation and their stable measurement performance. Thefrequency-domain algorithms use the complex band-pass filter with thepassband center at the fundamental frequency to extract the fundamentalphasor. The design methods of the complex band-pass filter can bedivided into two categories. One is based on Taylor Fourier Transform(TFT). This method uses Taylor series to approximate the time-varyingamplitude and phase, and then the weighted least square (WLS) method isused to solve the coefficients of Taylor series to obtain thecoefficients of complex band-pass filter. However, the TFT filteringcharacteristics are only determined by the data window length, the orderof Taylor series, and the weight coefficient, leading to its pooradjustability. In addition, the TFT has a poor anti-interferencecapability. To this end, harmonic and inter-harmonic signals can beincluded in the TFT model, but the number of interference signals mustbe estimated in advance, which increases the computational burden.Therefore, using the TFT method to estimate synchrophasor may not beideal in some circumstances.

The other design method is based on discrete Fourier Transform (DFT).This type of method designs a finite impulse response (FIR) lowpassfilter according to the standard requirements, and then uses the DFT toobtain the coefficients of the complex band-pass filter. The DFT basedmethod is flexible and applicable. The window function design iscommonly used to design lowpass filters, such as hamming, Taylor, andflat-top window. However, this method is not the optimal method forfilter design, and it cannot customize the gain of frequency bands. Assuch, it is difficult to effectively suppress the negative fundamentalcomponent.

Therefore, the estimation performance of the frequency-domain methods isdetermined by the filtering characteristics of the complex band-passfilters. To facilitate the design of filters and reduce trial and error,the filter design criteria may specify the range (e.g., stopband andpassband gain) of filter parameters. The filter parameter range withtotal vector error (TVE) limitations may be analyzed, and the influenceof frequency error (FE) and rate-of-change-of-frequency (ROCOF) error(RFE) limitations on the filter parameter range can be considered.Additionally, the influence of frequency deviation may be considered inamplitude modulation (AM) and phase modulation (PM) tests, and thedesign criteria. Also, the influence of frequency and ROCOF measurementalgorithm on the filter parameter range can be considered in OOB tests.In addition, the influence of the suppression of the negativefundamental component on the filter parameter range may be analyzed.Furthermore, some PMU standards have specific requirements for amplitudeerror (AE) and phase error (PE). The filter parameter range under may beconsidered provided AE and PE limitations.

In one example embodiment, the complex band-pass filter is selected todesign phasor algorithms. One has to determine the filter parameterrange to design the complex band-pass filter. To address theabove-mentioned problems, a general design method for phasor estimationin different scenarios is described. In particular, based on theabsolute value inequality and error limitation distribution, therequirements of passband and stopband gain of the complex band-passfilter are determined separately. Then, the general error models betweenerror limitations (TVE, AE, PE, FE, and RFE) and filter passband andstopband gain are constructed to determine the filter parameter range.

Moreover, the influence of the negative fundamental component, multipleinterference components, frequency deviation under modulationconditions, and the differentiators-based frequency and ROCOF estimationalgorithm on the filter parameter range is considered in the errormodels to enhance their applicability. According to the framework of thedesign method described in this disclosure, three classes of phasorestimation methods are designed, and their estimation performance isevaluated by experimental tests and compared with other estimationmethods.

PMU Estimation Models

A. Phasor Estimation

The power signal can be expressed as follows:y(t)=x(t)+η(t)=√{square root over (2)}a(t)cos(φ(t))+η(t)  (1)where x(t) is the fundamental signal; a(t) and φ(t) are the time-varyingamplitude and phase, respectively; η(t) is the interference signals(e.g., harmonics and OOB inter-harmonics).

Based on Euler's formula, the fundamental signal x(t) in (1) can berewritten as:

$\begin{matrix}{{x(t)} = {{\frac{\sqrt{2}}{2}{{a(t)}\left\lbrack {e^{j\;\varphi\;{(t)}} + e^{{- j}\;{\varphi{(t)}}}} \right\rbrack}} = {\frac{\sqrt{2}}{2}\left\lbrack {{X^{+}(t)} + {X^{-}(t)}} \right\rbrack}}} & (2)\end{matrix}$

Equation (2) indicates that the fundamental signal is composed ofsymmetrical positive (X⁺(t)) and negative (X⁻(t)) frequency componentsin the frequency domain.

According to the definition of a synchrophasor, the fundamental phasorof the power signal in (1) isX ^(&)(t)=a(t)e ^(jφ(t)) e ^(−j2πf) ^(n) ^(t) =X ⁺(t)e ^(−j2πf) ^(n)^(t)  (3)where X^(&) represents the synchrophasor to be reported, and f_(n)denotes the nominal frequency (f_(n)=50 Hz in this example, and thuse^(−j2πfnt) can be determined). Therefore, if X⁺(t) is estimated, thesynchrophasor can thereby be obtained.

Since the frequency components of X⁺(t) are mainly concentrated in thefrequency band near f_(n) (called measurement bandwidth), a phasorestimation algorithm can be designed as a band-pass digital filter toextract X⁺(t). The passband of the band-pass filter must contain themeasurement band to ensure the accurate extraction of X⁻(t). Inaddition, to eliminate the influence of η(t) on the phasor accuracy andavoid frequency aliasing during reporting the phasors, the band-passfilter must remove harmonics and OOB inter-harmonics. Thus, the stopbandrange of the band-pass filter must be lower than f_(n)−F_(r)/2 and overf_(n)+F_(r)/2 (F_(r) is the reporting rate). It should be noted thatX⁻(t) is also included in the stopband and its magnitude is far largerthan that of harmonics or OOB inter-harmonics. To ensure phasoraccuracy, the filter gain near X⁻(t) must be less than that of otherstopbands.

Let the FIR band-pass filter coefficients be h(k) (0≤k≤2N, where 2N isthe filter order), and then the positive fundamental component can beobtained by:

$\begin{matrix}{{z(k)} = {\sum\limits_{i = 0}^{2N}{{h(i)}{y\left( {k - N + i} \right)}}}} & (4)\end{matrix}$where y(k) is the discrete power signal, and z(k) is the measuredpositive fundamental component. The time stamp is marked in the middleof the data window to eliminate the phase shift. It should be noted thaty(k) is a real number and z(k) is a complex number; therefore, h(k) is acomplex band-pass filter.

Then, the synchrophasor at the report time can be obtained according tothe definition of the synchrophasor in (3):{dot over (X)}(k)=z(k)e ^(−j2πf) ^(n) ^(t) ^(k)   (5)where {dot over (X)}(k) is the discrete phasor and t_(k) is the reporttime.

B. Frequency and ROCOF Estimation

The frequency and ROCOF estimation methods can be divided into twotypes. One is based on the first derivative and second derivative of thephasor, which may be used in the TFT based methods. For this method, theinfluence of the FE and RFE limitations on the parameter range of h(k)can be analyzed. Another estimation method is based on differentiatorsthat can be obtained by difference method or LS method. For example, thefrequency estimation can be expressed as:

$\begin{matrix}{{f(k)} = {{\frac{1}{2\;\pi}{\sum\limits_{i = 0}^{2N_{d}}{{d(i)}{\varphi\left( {i - N_{d} + k} \right)}}}} + f_{n}}} & (6)\end{matrix}$where d(i) is the differentiator coefficients, φ(i) is the estimatedphase angle, and 2N_(d) is the differentiator order.

The differentiator can be regarded as a FIR filter. Its filteringcharacteristics affect the design of the complex band-pass filter h(k).In an example embodiment, the differentiator can be used to estimate thefrequency and ROCOF, and its influence on the parameter range of h(k)may be analyzed.

Framework for a Design Method

FIG. 1 shows an example framework for a design method for phasorestimation based on complex band-pass filters. In one exampleembodiment, based on the measurement requirements of applications andstandards, the range of filter parameters can be determined bymathematical error models. The complex band-pass filter is designed toestimate the synchrophasor based on the obtained filter parameter rangeafter checking the response time and latency.

In one example embodiment, in step 1, the measurement requirements canbe obtained. Different application scenarios have different requirementsthat include reporting rate, measurement bandwidth, and errorlimitations under static and dynamic conditions. For example, M-classand P-class PMU have different measurement bandwidths, which are 45Hz-55 Hz and 48 Hz-52 Hz, respectively. Before designing a phasoralgorithm, these requirements must be determined in advance.

In one example embodiment, in step 2, the filter parameter range may becalculated. To design a complex band-pass filter, one has to determineits parameters. The main filter parameters include the passband range,passband gain, stopband range, and stopband gain. For different classesof PMUs, the range of the passband and stopband can be obtained directlyby the measurement bandwidth requirement of different scenarios andreporting rate. The gain range of the passband and stopband can bedetermined by error limitations. Thus, error models between the filtergain and error limitations are analyzed, which determine the filter gainrange.

In step 3, the filter can be designed. Based on the range of filterparameters, the complex band-pass filter can be designed using any ofthe following techniques. For example, the WLS algorithm, iterativereweighting WLS (IRWLS) algorithm, or other similar algorithms may beused to design the complex band-pass filter. As another example, reallow-pass filters are designed first, and subsequently, complex band-passfilters are obtained using the DFT method.

In one example embodiment, the second design method can be used inphasor estimation, such as the method recommended in IEEE standard. Inanother example embodiment, the first method is the optimal designmethod and can customize the gain at different frequency bands. Inaddition, the first method may have better filtering characteristicsthan the second method under the same filter order. Therefore, the IRWLSalgorithm in the first method may be applied to design the complexband-pass filter.

In one example embodiment, in step 4, the response time and reportinglatency may be checked. After designing a complex band-pass filter, itis necessary to check whether its time response and latency depending ondata window meet the application requirements. If not, the complexband-pass filter may be redesigned after adjusting related requirements.

In Step 2, although the range of passband and stopband can be determinedby the reporting rate and the measurement bandwidth of differentapplications, it may be challenging to determine the range of passbandand stopband gain. Therefore, establishing the error models between theerror limits and filter gain ranges is an important aspect of thisdisclosure.

The current filter design criteria have various shortcomings. Forexample, the influence of amplitude and phase error limitations on thegain range is not analyzed. Additionally, the criteria neglect theinfluence of the frequency and ROCOF estimation method based ondifferentiator on the gain range. Moreover, the influence of multipleinterferences (e.g. negative fundamental component and OOBinter-harmonic) on the stopband gain range is not considered. To addressthese problems, this disclosure provides for general error modelsbetween the gain range and error limitations.

In the described framework, because the filter parameter range has beendetermined by error models, the designed h(k) can be used for phasorestimation directly as long as it meets the requirements of responsetime and reporting latency. However, if the filter parameter range isnot determined first, complete static and dynamic tests are required toverify whether the measurement performance of the designed filter meetsthe requirements. If not, the filter needs to be redesigned. The designprocess of phasor algorithms is tedious and time-consuming. Therefore,the proposed error models can facilitate the design of the complexband-pass filter and reduce trial and error.

General Error Models

Two inequalities will be used in the following analysis:

Inequality 1: Any two numbers a and b have the following inequalityrelations:∥a|−|b∥≤|a±b|≤|a|+|b|  (7)

Inequality 2: For any real number θ, the arctangent function has thefollowing relations:|arctan θ|=arctan|θ|≤|θ|  (8)

In addition, if |θ|≤2°, |θ−arctan θ|≤0.00082°. Thus, if θ is small,arctan θ≈θ.

FIG. 2 shows an example diagram of phasor estimation filters designed byIRWLS algorithm. In this example embodiment, |H(f)| represents themagnitude response of h(k), δ is the passband ripple, indicating thatthe passband gain fluctuates between 1−δ and 1+δ, and the stopband gainis less than λ. In this section, the task is to establish the errormodels between the filter gain and application requirements. Based onthe models, the maximum values of δ and λ can be obtained to set upcriteria for the design of h(k).

When using h(k) to measure the phasor, the estimation error may fulfillthe following equation:E=|E(δ)+E(λ)|≤E _(lim)  (9)where E is the estimation error, E(δ) is the error caused by the rippleδ, E(λ) is the error resulting from the interference components that arenot fully suppressed, and E_(lim) is the error limitation from thestandards or application requirements. There is a coupling relationshipbetween the errors caused by δ and λ. In such case, it is difficult toanalyze the range of δ and λ. However, according to Inequality 1, theerrors caused by δ and λ can be obtained by:E≤|E(δ)|+|E(λ)|≤E _(lim)  (10)where the errors caused by δ and λ are independent of each other. Thus,the analysis of the passband ripple δ is not involved in the stopbandgain 2, and vice versa. To meet the requirements, E(δ) and E(λ) mustsatisfy the following equation:|E(δ)|≤E _(pass) , |E(λ)|≤E _(stop) , E _(pass) +E _(stop) ≤E_(lim)  (11)where E_(pass) and E_(stop) are the given limitations of errors causedby δ and λ, and their sum must be smaller than E_(lim). In practicalapplications, hardware uncertainty and random noise may exist. To meetthe requirements, E_(pass) and E_(stop) must have a certain margin.Static signals have more accuracy requirements and are more sensitive torandom noise. Therefore, in one example embodiment,E_(pass)=E_(stop)=E_(lim)/10 for static conditions andE_(pass)=E_(stop)=E_(lim)/3 for dynamic conditions. Other errorlimitation distributions satisfying (11) are also allowed. Therefore,the range of the passband and stopband gain can be analyzed separately.

Based on (11), we want to use general error models to obtain the gainrange:δ≤S ₁(E _(pass)), λ≤S ₂(E _(stop))  (12)where S₁(●) and S₂(●) are the functions with E_(pass) and E_(stop),respectively. Equation (12) indicates that the range of δ and λ can bedetermined as long as the error limitations of the passband and stopbandare given.

There are three kinds of errors for the application requirements, namelyphasor error, FE, and RFE. In IEEE standards, TVE is used to evaluatethe phasor accuracy. However, AE and PE are used to evaluate the phasoraccuracy in some standards, such as Chinese PMU standards. Thus, to makeerror models more general, there are five kinds of error limitations:TVE_(pass) (TVE_(stop)), AE_(pass) (AE_(stop)), PE_(pass) (PE_(stop)),FE_(pass) (FE_(stop)), and RFE_(pass) (RFE_(stop)). The range of δ and λis analyzed based on these error limitations.

A. Error Models for Passband Gain

The ideal passband gain is one, but in practical applications thepassband gain of h(k) has fluctuation and attenuation. Therefore, δneeds to be limited to ensure the phasor accuracy. Due to the lengthlimitation, the PM signal, which is the most complex scenario, is takenas an example to deduce the general error models and the range of δ.

When the phase angle is sinusoidally modulated, the power signal modelcan be modelled as follows:x(t)=√{square root over (2)}X _(m) cos[2πf ₀ t+k _(p) sin(2πf _(m) t+φ_(m))+φ₀]  (13)where X_(m), f₀ and φ₀ are the amplitude, frequency, and initial phaseof the fundamental signal, respectively; k_(a), f_(m) and φ_(m) are thephase modulation depth, frequency and initial phase, respectively.According to the Bessel function, the above equation can be decomposedas follows:

$\begin{matrix}{{x(t)} = {\sqrt{2}X_{m}{\sum\limits_{n = {- \infty}}^{\infty}{{J_{n}\left( k_{p} \right)}\left( {- 1} \right)^{n}{\cos\;\left\lbrack {{2\;{\pi\left( {f_{0} + {nf}_{m}} \right)}t} + \varphi_{0} + {n\;\varphi_{m}}} \right\rbrack}}}}} & (14)\end{matrix}$where J_(n)(●) is the first Bessel function of n order.

The f₀ and f₀±f_(m) components are in the passband of h(k) and theirestimation errors are denoted as ΔE₁. In addition, the f₀±2f_(m),f₀±3f_(m) and other components also affect the estimation accuracy. Whenthe reporting rate is small and the modulation frequency is large (e.g.,=25 Hz and f_(m)=5 Hz), the phasor estimation methods may suppress thesefrequency components due to its narrow passband for effectively reducingOOB inter-harmonics. This may cause an error of ΔE₂, which is notconsidered in other methods.

The worst case is that other components are completely filtered out, andonly f₀ and f₀±f_(m) remained. In this case, |ΔE₂| is the largest andthe acceptable Δ|E₁| is the smallest according to (15). Once δ can meetthe requirements under this scenario, it is also suitable for otherF_(r) and f_(m) values. Therefore, the following analysis focuses onthis most stringent case.E(δ)=|ΔE ₁ +ΔE ₂ |≤|ΔE ₁ |+|ΔE ₂ |≤E _(pass)  (15)

When there are only the three components f₀ and f₀±f_(m), the extractedphasor can be represented as follows:

$\begin{matrix}\begin{matrix}{{z(t)} = {X_{m}\left\lbrack {{{J_{0}\left( k_{p} \right)}{{H\left( f_{0} \right)}}e^{j\; 2\;\pi\; f_{0}t}} + {{J_{1}\left( k_{p} \right)}{{H\left( {f_{0} + f_{m}} \right)}}e^{j\; 2\;{\pi{({f_{0} + f_{m}})}}t}} -} \right.}} \\\left. {{J_{1}\left( k_{p} \right)}{{H\left( {f_{0} - f_{m}} \right)}}e^{j\; 2\;{\pi{({f_{0} - f_{m}})}}t}} \right\rbrack \\{= {{X_{m}{e^{j\; 2\;\pi\; f_{0}t}\left( {k_{p\; 0} + {k_{p\; 1}e^{j\; 2\;\pi\; f_{m}t}} - {k_{p\; 2}e^{{- j}\; 2\pi\; f_{m}t}}} \right)}} = {X_{m}e^{j\; 2\;\pi\; f_{0}t}\Delta\;{z(t)}}}}\end{matrix} & (16)\end{matrix}$where the initial phase is set as zero for convenience of thederivative, and additionally k_(p0)=J₀(k_(p))|H(f₀)|,k_(p1)=J₁(k_(p))|H(f₀+f_(m))|, and k_(p2)=J₁(k_(p))|H(f₀−f_(m))|. The f₀may not be the nominal frequency. If the frequency-tracking technique isapplied, k_(p1) may be equal to k_(p2). But for fixed filters,k_(p1)≠k_(p2), which is more general. If the error models fork_(p1)≠k_(p2) is established, those may be also suitable fork_(p1)=k_(p2). Therefore, k_(p1)≠k_(p2) is considered in the followinganalysis, which is not involved in the existing filter design criteria.

The theoretical phasor is:X ⁺(t)=X _(m) e ^(j(2πf) ⁰ ^(t+k) ^(p) ^(sin 2πf) ^(m) ^(t))  (17)

i. Phasor Error Limitation

As mentioned above, different standards have different evaluationcriteria for phasor accuracy. TVE, AE, and PE are commonly used.

a) Amplitude Error Limitation

The amplitude error isAE=∥X ⁺(t)|−|z(t)∥/|X ⁺(t)|=|1−|Δz(t)∥  (18)

Let k_(p2)=k_(p1)+Δk_(p2), and then

$\begin{matrix}\begin{matrix}{{\Delta\;{z(t)}} = {k_{p\; 0} + {k_{p\; 1}e^{j\; 2\;\pi\; f_{m}t}} - {k_{p\; 2}e^{{- j}\; 2\;\pi\; f_{m}t}}}} \\{= {k_{p\; 0} + {k_{p\; 1}e^{j\; 2\;\pi\; f_{m}t}} - {\left( {k_{p\; 1} + {\Delta\; k_{p\; 2}}} \right)e^{{- j}\; 2\;\pi\; f_{m}t}}}} \\{= {k_{p\; 0} + {j\; 2\; k_{p\; 1}\sin\; 2\;\pi\; f_{m}t} - {\Delta\; k_{p\; 2}e^{{- j}\; 2\;\pi\; f_{m}t}}}}\end{matrix} & (19)\end{matrix}$

After analysis on (19), the range of |Δz(i)| is:k _(p0) −|Δk _(p2) |≤|Δz(t)|≤√{square root over (k _(p0) ²+4k _(p1)²)}+|Δk _(p2)|  (20)

Thus, the range of AE has two cases:AE≤|1−k _(p0) +|Δk _(p2)∥≤|1−k _(p0) |+|Δk _(p2)|  (21)orAE≤|1−√{square root over (k _(p0) ²+4k _(p1) ²)}−|Δk _(p2)∥=√{squareroot over (k _(p0) ²+4k _(p1) ²)}+|Δk _(p2)|−1  (22)

The range of the passband gain is between 1−δ and 1+δ. Thus, the rangeof k_(p0) is between J₀(k_(p))(1−δ) and J₀(k_(p))(1−δ), and the range ofk_(p1) and k_(p2) is between J₁(k_(p))(1−δ) and J₁(k_(p))(1−δ). Afteranalysis on (21) and (22), the maximum value of AE is:AE≤1−(1−δ)J ₀(k _(p))+2δ·J ₁(k _(p))≤AE _(pass)  (23)orAE=√{square root over (J ₀(k _(p))²+4J ₁(k _(p))²)}·(1+δ)+2δ·J ₁(k_(p))−1≤AE _(pass)  (24)

Then, the range of the ripple δ can be obtained:δ≤(AE _(pass)−1+J ₀(k _(p)))/(J ₀(k _(p))+2J ₁(k _(p)))  (25)or

$\begin{matrix}{\delta \leq \frac{{AE}_{pass} + 1 - \sqrt{{J_{0}\left( k_{p} \right)}^{2} + {4{J_{1}\left( k_{p} \right)}^{2}}}}{\sqrt{{J_{0}\left( k_{p} \right)}^{2} + {4{J_{1}\left( k_{p} \right)}^{2}} + {2{J_{1}\left( k_{p} \right)}}}}} & (26)\end{matrix}$δ must be set as the minimum range of (25) and (26).

b) Phase Error Limitation

Since |Δk_(p2)<<√{square root over (k_(p0) ²+4k_(p1) ² sin² 2πf_(m)t)}in Δz(t), the phase angle can be approximated as (using Inequality 2)

$\begin{matrix}\begin{matrix}{{{\angle\;\Delta\;{z(t)}} \approx {\angle\left( {k_{p\; 0} + {j\; 2k_{p\; 1}\sin\; 2\;\pi\; f_{m}t}} \right)}} = {\arctan\left( {2k_{p\; 1}\sin\; 2\;\pi\; f_{m}{t/k_{p\; 0}}} \right)}} \\{\approx {\left( {2{k_{p\; 1}/k_{p\; 0}}} \right)\sin\; 2\;\pi\; f_{m}t}}\end{matrix} & (27)\end{matrix}$

The phase error is then:

$\begin{matrix}{{PE} = {{{{\angle\;{X^{+}(t)}} - {\angle\;{z(t)}}}} \approx {{\left( {k_{p} - {2{k_{p\; 1}/k_{p\; 0}}}} \right)\sin\; 2\;\pi\; f_{m}t}} \leq {{k_{p} - \frac{2k_{p\; 1}}{k_{p\; 0}}}} \leq {{k_{p} - \frac{{J_{1}\left( k_{p} \right)}\left( {2 + {2\;\delta}} \right)}{{J_{0}\left( k_{p} \right)}\left( {1 - \delta} \right)}}} \leq \;{PE}_{pass}}} & (28)\end{matrix}$

Therefore, the ripple δ has the following range:

$\begin{matrix}{\delta \leq \frac{{\left( {{PE}_{pass} + k_{p}} \right){J_{0}\left( k_{p} \right)}} - {2\;{J_{1}\left( k_{p} \right)}}}{{\left( {{PE}_{pass} + k_{p}} \right){J_{0}\left( k_{p} \right)}} + {2{J_{1}\left( k_{p} \right)}}}} & (29)\end{matrix}$

c) TVE Limitation

Similar to the above analysis, the range of δ under TVE limitation isgiven directly:δ≤(TVE_(pass)−TVE₂)/(K ₀(k _(p))+2J ₁(k _(p)))  (30)where TVE₂ is independent of the passband gain and depends on f₀±2f_(m),f₀±3f_(m) and other components, and it can be calculated by simulation.

ii. Frequency Error Limitation

According to (28), the frequency error can be obtained by:

$\begin{matrix}{{FE} = {{\frac{1}{2\;\pi} \cdot \frac{d({PE})}{dt}} \approx {\frac{1}{2\;\pi}{{\left( {k_{p} - {2{k_{p\; 1}/k_{p\; 0}}}} \right)2\;\pi\; f_{m}\cos\; 2\;\pi\; f_{m}t}}}}} & (31)\end{matrix}$

In practical applications, the numerical differentiation is used toreplace the derivative. In this disclosure, the differentiator is usedto estimate the frequency, and its magnitude response is denoted asD(f). Therefore, the derivative of sin(2πf_(m)t) must beD(f_(m))·cos(2πf_(m)t), instead of (2πf_(m))·cos(2πf_(m)t). The aboveequation must be modified as:

$\begin{matrix}{{FE} = {{\frac{1}{2\;\pi}{{\left( {k_{p} - {2{k_{p\; 1}/k_{p\; 0}}}} \right){D\left( f_{m} \right)}\cos\; 2\;\pi\; f_{m}t}}} \leq {{{PE} \cdot {{D\left( f_{m} \right)}/2}}\;\pi} \leq \;{FE}_{pass}}} & (32)\end{matrix}$

Then, the range of δ is:

$\begin{matrix}{\delta \leq \frac{{\left( {{2{\pi \cdot {{FE}_{pass}/{D\left( f_{m} \right)}}}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} - {2{J_{1}\left( k_{p} \right)}}}{{\left( {{2{\pi \cdot {{FE}_{pass}/{D\left( f_{m} \right)}}}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} - {2{J_{1}\left( k_{p} \right)}}}} & (33)\end{matrix}$

iii. ROCOF Error Limitation

Like the frequency estimation, the ROCOF error is:

$\begin{matrix}{{RFE} = {{\frac{1}{2\;\pi} \cdot \frac{d^{2}({PE})}{{dt}^{2}}} \leq {\frac{1}{2\;\pi}{{PE} \cdot {R\left( f_{m} \right)}}} \leq {RFE}_{pass}}} & (34)\end{matrix}$where R(f) is the magnitude response of the differentiator to estimatethe ROCOF. Thus, the ripple δ must meet:

$\begin{matrix}{\delta \leq \frac{{\left( {{2{\pi \cdot {{RFE}_{pass}/{R\left( f_{m} \right)}}}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} - {2{J_{1}\left( k_{p} \right)}}}{{\left( {{2{\pi \cdot {{RFE}_{pass}/{R\left( f_{m} \right)}}}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} - {2{J_{1}\left( k_{p} \right)}}}} & (35)\end{matrix}$

The analysis process for the error models of other signal types includesthe same steps as those from 1) to 3), and thus no additional detailsare given.

B. Error Models for Stopband Gain

The stopband of h(k) is used to suppress interference signals, includingXV), harmonics and OOB inter-harmonics. The existing design criteria haspaid no attention to the influence of the negative fundamentalcomponent.

i. Error Models for the Negative Fundamental Component

When there is no OOB and harmonic in the signal, only X⁻(t) needs to berejected. In this case, the extracted phasor is:z(t)=X _(m) e ^(j2πf) ⁰ ^(t) +|H(−f ₀)|X _(m) e ^(−j2πf) ⁰ ^(t)  (36)

For the sake of generality, the above equation is rewritten as:

$\begin{matrix}\begin{matrix}{{z(t)} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( f_{i} \right)}}k_{i}X_{m}e^{j\; 2\;\pi\; f_{i}t}}}} \\{= {X_{m}{e^{j\; 2\;\pi\; f_{0}t}\left( {1 + {{{H\left( f_{i} \right)}}k_{i}e^{j\; 2\;\pi\;{({f_{i} - f_{0}})}t}}} \right)}}}\end{matrix} & (37)\end{matrix}$where f_(i) is the frequency of one interference signal, k_(i) is theratio of the magnitude of the interference signal to the fundamentalmagnitude. In particular, if f_(i)=−f₀, then k_(i)=1.

Let G_(i)=|H(f_(i))·k_(i) and p_(i)=2π(f_(i)−f₀), thenΔz(t)=1+|h(f _(i))|k _(i) e ^(j2π(f) ^(i) ^(-f) ⁰ ^()t)=1G _(i)+cos p_(i) t+jG _(i) sin p _(i) t  (38)

a) Phasor Error Limitation

(1) Amplitude Error Limitation

The amplitude error is:

$\begin{matrix}\begin{matrix}{{AE} = {\frac{{{{X^{+}(t)}} - {{z(t)}}}}{{X^{+}(t)}} = \frac{{X_{m} - {X_{m}\sqrt{1 + {2G_{i}\cos\; p_{i}t} + G_{i}^{2}}}}}{X_{m}}}} \\{= {{{1 - \sqrt{1 + {2G_{i}\cos\; p_{i}t} + G_{i}^{2}}}} \leq G_{i} \leq {AE}_{stop}}}\end{matrix} & (39)\end{matrix}$

Thus, the stopband gain can be obtained as follows:|H(f _(i))|≤AE _(stop) /k _(i)  (40)

(2) Phase Error Limitation

PE is (the maximum value is taken when cos(p_(i)t)=−G_(i))

$\begin{matrix}{{PE} = {{{\angle\;\Delta\;{z(t)}}} = {{{{arc}\;\tan\frac{G_{i}\sin\; p_{i}t}{1 + {G_{i}\cos\; p_{i}t}}}} \leq {{arc}\;\tan\frac{G_{i}}{\sqrt{1 - G_{i}^{2}}}} \leq {PE}_{stop}}}} & (41)\end{matrix}$

Thus, the stopband gain must satisfy:

$\begin{matrix}{{{H\left( f_{i} \right)}} \leq {\frac{1}{k_{i}}\frac{\tan\;{PE}_{stop}}{\sqrt{1 + {\tan^{2}\;{PE}_{stop}}}}}} & (42)\end{matrix}$

(3) TVE Limitation

The phasor error is:TVE=|X ⁺(t)−z(t)|/|X ⁺(t)|=G _(i)≤TVE_(stop)  (43)

Thus, the stopband gain has the following range:|H(f _(i))|≤TVE_(stop) /k _(i)  (44)

b) Frequency Error Limitation

The frequency error is:

$\begin{matrix}{{FE} = {{\frac{1}{2\;\pi}{\frac{d\;\angle\;\Delta\;{z(t)}}{d\; t}}} = {\frac{{G_{i}p_{i}\cos\; p_{i}t} + {G_{i}^{2}p_{i}}}{1 + {2G_{i}\cos\; p_{i}t} + G_{i}^{2}}}}} & (45)\end{matrix}$

In practical applications, the numerical differentiation is used toreplace the derivation. The differentiator is used to estimate thefrequency and ROCOF, and its magnitude response is denoted as D(f).Therefore, the derivative expression of sin(p_(i)t) must beD(f_(i)−f₀)·cos(p_(i)t) instead of p_(i)·cos(p_(i)t). The above equationmust be rewritten as:

$\begin{matrix}\begin{matrix}{{FE} = {\frac{1}{2\;\pi}{\frac{{G_{i}{D\left( {f_{i} - f_{0}} \right)}\cos\; p_{i}t} + {G_{i}^{2}{D\left( {f_{i} - f_{0}} \right)}}}{1 + {2G_{i}\cos\; p_{i}t} + G_{i}^{2}}}}} \\{\leq {\frac{1}{2\;\pi}{\frac{G_{i}{D\left( {f_{i} - f_{0}} \right)}}{1 + G_{i}}}} \leq {FE}_{stop}}\end{matrix} & (46)\end{matrix}$

Therefore, the range of the stopband gain is:

$\begin{matrix}{{{H\left( f_{i} \right)}} \leq {\frac{1}{k_{i}}\frac{2\;{\pi \cdot {{FE}_{stop}/{D\left( {f_{i} - f_{0}} \right)}}}}{1 + {2\;{\pi\; \cdot {{FE}_{stop}/{D\left( {f_{i} - f_{0}} \right)}}}}}}} & (47)\end{matrix}$

c) ROCOF Error Limitation

The analysis process is the same as the above b), and it is notdiscussed in detail here. For the given ROCOF error limitation, therange of the stopband gain can be obtained as:

$\begin{matrix}{{{H\left( f_{i} \right)}} \leq {\frac{1}{k_{i}}\frac{2\;{\pi \cdot {{RFE}_{stop}/{R\left( {f_{i} - f_{0}} \right)}}}}{\sqrt{1 + \left( {2\;{\pi\; \cdot {{RFE}_{stop}/{R\left( {f_{i} - f_{0}} \right)}}}} \right)^{2}}}}} & (48)\end{matrix}$

ii. Error Models for OOB or Harmonic

When there is a single harmonic or OOB inter-harmonic in the powersignal, its positive and negative components exist in the frequencydomain according to Euler's formula. In addition, X⁻(t) is also includedin the power signal. This means that three interference components needto be rejected at the same time. On this condition, the phasor obtainedis as follows:

$\begin{matrix}\begin{matrix}{{z(t)} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( {- f_{0}} \right)}}X_{m}e^{{- j}\; 2\;\pi\; f_{0}t}} +}} \\{{{{H\left( f_{i} \right)}}k_{i}X_{m}e^{j\; 2\;\pi\; f_{i}t}} + {{{H\left( {- f_{i}} \right)}}k_{i}X_{m}e^{{- j}\; 2\;\pi\; f_{i}t}}} \\{= {\left\lbrack {{X_{m}e^{j\; 2\pi\; f_{0}t}} + {{{H\left( {- f_{0}} \right)}}X_{m}e^{{- j}\; 2\;\pi\; f_{0}t}}} \right\rbrack +}} \\{\left\lbrack {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( f_{i} \right)}}k_{i}X_{m}e^{j\; 2\;\pi\; f_{i}t}}} \right\rbrack -} \\{\left\lbrack {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} - {{{H\left( {- f_{i}} \right)}}k_{i}X_{m}e^{{- j}\; 2\;\pi\; f_{i}t}}} \right\rbrack} \\{= {{z_{1}(t)} + {z_{2}(t)} - {z_{3}(t)}}}\end{matrix} & (49)\end{matrix}$

The theoretical phasor is:

$\begin{matrix}\begin{matrix}{{X^{+}(t)} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {X_{m}e^{j\; 2\pi\; f_{0}t}} - {X_{m}e^{j\; 2\;\pi\; f_{0}t}}}}} \\{= {{X_{1}^{+}(t)} + {X_{2}^{+}(t)} - {X_{3}^{+}(t)}}}\end{matrix} & (50)\end{matrix}$

E(X⁺(t), z(t)) is denoted as the estimation error between X⁺(t) andz(t), and thus the overall error is:

$\begin{matrix}{{E(\lambda)} = {{{{E\left( {{X_{1}^{+}(t)},{z_{1}(t)}} \right)} + {E\left( {{X_{2}^{+}(t)},{z_{2}(t)}} \right)} - {E\left( {{X_{3}^{+}(t)},{z_{3}(t)}} \right)}}} \leq {{{E\left( {{X_{1}^{+}(t)},{z_{1}(t)}} \right)}} + {{E\left( {{X_{2}^{+}(t)},{z_{2}(t)}} \right)}} + {{E\left( {{X_{3}^{+}(t)},{z_{3}(t)}} \right)}}} \leq E_{stop}}} & (51)\end{matrix}$

The estimation error of multiple components is smaller than the sum ofthe errors of each component. The error analysis of multiple componentsis transformed into that of a single component. To meet therequirements, the error limitations of each part is set as E_(stop)/3.The detailed analysis is the same as that described in the abovesubsection, and is not discussed here.

C. Design Procedure for Phasor Estimation Method

In one example embodiment, the phasor estimation methods of differentclasses of PMUs can be designed as follows:

First, the measurement passband range, stopband range, and errorlimitations are determined according to different standards orapplication requirements. Second, the range of the passband and stopbandgain is obtained by substituting the error limitations into the errormodels, such as (29), (30), (33), (42), (44), and (47). Different signaltypes may have different δ and λ, and then the minimum values of allobtained δ and λ are selected as the final gain range.

Third, according to the above parameters, the IRWLS algorithm is used todesign the complex band-pass filter.

Fourth, based on the above analysis, the phasor estimation filter h(k)can be used to develop the phasor algorithm, and then the synchrophasorcan be estimated.

Case Study

A. Design of Different Classes of Phasor Estimation Methods

According to the above procedure, different classes of phasor estimationmethods (M-class, P-class, and S-class) can be designed.

FIG. 3 shows example filtering characteristics and parameter ranges forM-class, P-class, and S-class filters. Additionally, Table I showsexample filtering characteristics and parameter ranges for M-class,P-class, and S-class filters. The S-class PMU is proposed to monitor theSSO in real-time, because of the SSO affecting the stability of thepower systems. When SSOs occur in the power systems, the power signalsmay have high-frequency oscillations of up to 30 Hz or higher. However,M-class PMUs focus on estimating the power signals with an oscillationfrequency of less than 5 Hz. Therefore, a phasor estimation method forthe S-class PMU is described to measure the phasor oscillations at highfrequency.

During the design process, the reporting rates of the M- and S-classPMUs are 100 Hz, so it is necessary to filter out the OOB interferencesignals over 100 Hz. For the P-class PMU, its stopband only needs tofilter out the harmonic signals and does not need to have thesuppression ability of the OOB signals. The algorithm of the S-class PMUis designed to monitor SSO of 30 Hz, so the passband range is 20-80 Hz.The passband range and error limitations of the M-class and P-class PMUare known. There are no standards for the S-class PMU, so the errorlimitations of the M-class PMU are used to determine the gain range ofthe S-class PMU in this disclosure.

The “Required” in Table I means that the gain ranges of passband andstopband are calculated according to the described general error model.The “Designed” is the real gains of the designed filters. According tothe above analysis, the magnitude of the negative fundamental componentX⁻(X) is more than that of the harmonic and OOB inter-harmonic. Thus,the gain range of X⁻(t) is less than that of other stopbands. The“Designed” gains are in the range of the “required” gains, and thus theestimation accuracy of the phasor methods can meet the standard andapplication requirements.

The step response and reporting latency of the designed algorithms havebeen checked, and they meet the standard requirements. Therefore,different classes of methods can be used for the phasor estimation indifferent application scenarios.

TABLE I MEASUREMENT REQUIREMENTS AND FILTER PARAMETER RANGE OF DIFFERENTCLASSES OF PHASOR ESTIMATION METHODS M-class P-class S-class Passbandrange 45-55 Hz 48-52 Hz 30-80 Hz Stopband range ≥100 Hz Harmonic ≥100 HzPassband Required (dB) <0.0087 0.059 <0.0087 ripple Designed (dB) 0.0070.035 0.005 Stopband X⁻(t) Required (dB) <−89 <−79 <−89 gain Designed(dB) −112 −110 −115 Harmonic/ Required (dB) <−67.6 <−59 <−67.6 OOBDesigned (dB) −69 −100 −68 Data window (cycles) 4 2 10

TABLE II MAXIMUM ERRORS OF M-CLASS PMUS UNDER STEADY-STATE AND DYNAMICCONDITIONS TVE (%) FE (Hz) RFE (Hz/s) Test types Std M_(A) M_(B) M_(C)Std M_(A) M_(B) M_(C) Std M_(A) M_(B) M_(C) Off-nominal 1 0.003 0.0030.133 0.005 1.2 × 10⁻⁴ 0.008 0.012 0.1 0.007 0.004 0.689 Harmonics/OOB 10.010 0.065 0.137 0.025 1.9 × 10⁻⁴ 0.003 0.004 — 0.033 0.127 0.216 AM 30.011 1.104 0.759 0.01 3.7 × 10⁻⁴ 0.008 0.008 14 0.014 0.203 0.209 PM 30.015 0.998 0.688 0.3 0.009 0.031 0.040 14 0.523 1.190 1.470 Frequencyramp 1 0.011 0.058 0.211 0.01 1.3 × 10⁻⁴ 0.009 0.012 0.2 0.012 0.0070.677

TABLE III RESPONSE TIME FOR M-CLASS PMUS Phasor (ms) Frequency (ms)ROCOF (ms) Step types Std M_(A) M_(B) M_(C) Std M_(A) M_(B) M_(C) StdM_(A) M_(B) M_(C) Amplitude step 140 21 40 32 280 75 62 69 280 106 86 94Phase step 140 21 46 37 280 87 64 73 280 111 87 112

B. Experimental Tests

The three phasor estimation methods were implemented in the PMUprototype. The hardware framework of the PMU prototype and its explainedbelow.

i. M-Class PMU

The test conditions for the M-class PMUs under static and dynamicconditions are explained in IEEE and Chinese PMU standards. It should benoted that the fundamental frequency offsets ±0.5 Hz in the harmonic,OOB, AM, and PM tests. The measurement accuracy of the proposed M-classphasor algorithm (M_(A)) is compared with that of theiterative-Interpolated DFT (i-IpDFT) algorithm (M_(B)) and the algorithmrecommended in IEEE Std. (M_(C)). For the sake of comparison, the datawindows are all 4 cycles. The test results are shown in Tables II andIII.

As shown in Table II, the described method M_(A) suppresses X⁻(t); thus,the accuracy is high in the off-nominal test. The M_(B) methodsuppresses X⁻(t) using an iterative technique, resulting in highaccuracy as well. However, the M_(C) method does not suppress X⁻(t)effectively, causing larger estimation errors. M_(A) provides goodsuppression performance for harmonics and OOB inter-harmonics based onthe error models. The iterative technique in M_(B) is used to reduce theinfluence of interferences, but the accuracy is lower than that ofM_(A). Compared with M_(A), the M_(C) method is not optimal and itexhibits poor performance for interference suppression. Under dynamicconditions, the passband ripple of M_(A) is small and X⁺(t) can beextracted accurately. However, M_(B) and M_(C) do not have the flat-toppassband gain and their errors are larger than that of M_(A). Theaccuracy of M_(A) is at least one order of magnitude better than thestandard requirements under static and dynamic conditions.

As shown in Table III, the proposed method M_(A) has the shortest phasorresponse time in the same data window. The data window of the frequencyand ROCOF estimation for M_(B) is one cycle less than those of M_(A) andM_(C); therefore, the frequency and ROCOF response time are theshortest. However, the response time of M_(A) is within the limitations.

ii. P-Class PMU

The test conditions are known in the art. The fundamental frequency alsooffsets ±0.5 Hz in the harmonic, AM, and PM tests, and the harmonicmagnitude is 10% of the fundamental magnitude. The P-class algorithm inIEEE Std. (P_(B)) and the AM model method (P_(C)) are compared with theproposed method (P_(A)). In addition, a correction method is applied toimprove the accuracy of P_(A), because of the larger passbandattenuation. The maximum errors and response times are shown in TablesIV and V.

As listed in Table IV, P_(C) provides accurate estimates for signalswithout any interference, but the errors are too large to meet thelimitations in harmonic tests. Many harmonics may occur in actual powersystems, especially in the renewable energy field. Thus, P_(C) may bedifficult to provide accurate phasors if it is not improved. Theaccuracies of P_(A) and P_(B) meet the requirements in all tests. In theoff-nominal and frequency ramp tests, P_(B) exhibits less suppression ofX⁻(t) than P_(A); thus, the phasor errors are larger. Because thedesigned P_(A) successfully filters out the harmonics, the measurementsare accurate, even for a 10% harmonic content. The correction techniquein IEEE Std. is only suitable for the frequency offset but not fordynamic modulation. However, the proposed correction method isapplicable to both conditions. Hence, the phasor errors of P_(B) arehigher than those of P_(A) in the AM and PM tests. In the step tests(Table V), the three methods exhibit small differences in the frequencyand ROCOF response time. However, the phasor response time of P_(A) isshortest due to the use of the proposed correction method. To sum up,the described method has good performance under static and dynamicconditions, and its accuracy is at least 10 times better than therequirements.

TABLE IV MAK uvium ERRORS OF P-CLASc PMUS UNDER STEADY-STATE AND DYNAMICCONDITIONS TVE (%) FE (Hz) RFE (Hz/s) Test types Std P_(A) P_(B) P_(C)Std P_(A) P_(B) P_(C) Std P_(A) P_(B) P_(C) Off-nominal 1 0.005 0.0440.004 0.005 1.2 × 10⁻⁴ 0.002 2.2 × 10⁻⁴ 0.4 0.009 0.057 0.095 Harmonics1 0.005 0.014 3.151 0.005 1.0 × 10⁻⁴ 1.5 × 10⁻⁴ 0.025 0.4 0.005 0.011142 AM 3 0.006 0.059 0.0011 0.06 5.9 × 10⁻⁴ 7.4 × 10⁻⁴ 3.2 × 10⁻⁴ 2.30.014 0.032 0.154 PM 3 0.006 0.064 0.009 0.06 0.002 0.002 7.4 × 10⁻⁴ 2.30.025 0.039 0.146 Frequency ramp 1 0.007 0.053 0.013 0.01 1.9 × 10⁻⁴0.002 2.6 × 10⁻⁴ 0.4 0,013 0.044 0.112

TABLE V RESPONSE TIME FOR P-CLASS PMUS Phasor (ms) Frequency (ms) ROCOF(ms) Step types Std P_(A) P_(B) P_(C) Std P_(A) P_(B) P_(C) Std P_(A)P_(B) P_(C) Amplitude step 40 15 23 18 90 51 53 60 120 58 56 60 Phasestep 40 19 28 33 90 56 58 59 120 59 58 60

iii. S-Class PMU

No requirements are specified for S-class PMUs in existing PMUstandards. The S-class PMU focuses on the phasor extraction ofhigh-frequency oscillation signals. Therefore, the following tests areused to evaluate the performance of the S-class PMU.

a) Modulation Tests

The power signals during SSOs are modulated at high frequency. Thus, AMand PM signals with high modulation frequency are used to test theaccuracy of the designed phasor algorithm. Under actual fieldconditions, the harmonics and OOB inter-harmonics may be superimposedonto the SSO signals, which may interfere with the monitoring of SSOs.In addition, the fundamental frequency may deviate from 50 Hz.Therefore, the AM or PM test signals contain the harmonics of 100 Hz and150 Hz and the OOB inter-harmonics of 110 Hz and 130 Hz. The magnitudesof these interferences are 10% of the fundamental magnitude and thefundamental frequency offsets ±1 Hz.

FIG. 4 displays the phasor errors of an example S-class PMU in the AMand PM tests. The blue curve (i.e., the curve on the top) shows thephasor errors of the test signals with the interferences and the redcurve (i.e., the curve on the bottom) denotes the phasor errors of theinterference-free signals. The red curve ranges between 0.007% and0.011%, and the range of the blue curve is 0.017% and 0.022% in the AMtests. In the PM tests, the values of the red curve increase from 0.005%to 0.33%, and those of the blue curve increase from 0.018% to 0.33%.According to the above analysis, PM signals can be decomposed intomultiple components. As the modulation frequency increases, f₀±2f_(m),f₀±3f_(m), and the other components may be suppressed, causing anincrease in phasor errors. This part of the error is considered in thegeneral error models; thus, the phasor accuracy is better than therequirements. In addition, the phasor errors increase due tointerferences. However, the increase in the errors is relatively small,and the phasor accuracy still satisfies the TVE requirement of 3% forM-class PMUs. The results indicate that the PMU prototype can extractthe phasor of the high-frequency oscillation signals and monitor the SSOsuccessfully.

b) Performance Comparison

The DFT and Taylor weighted least squares (TWLS) methods may be used asphasor estimators. Due to the disadvantages of these methods, someimprovements have been considered. In this disclosure, the Clarketransformation-based DFT algorithm (CT-DFT) and the TWLS-IpDFT arecompared with the proposed method.

FIG. 5 shows example phasor errors obtained from interference tests. Inparticular, FIG. 5 shows phasor errors of the CT-DFT (a) and TWLS-IpDFT(b) methods in the AM and PM tests with interference signals (the AMcurve is the curve on the top). The phasor errors of both methods riserapidly with an increase in the modulation frequency in the AM and PMtests. The maximum TVE is 11%; thus, these two methods are not suitablefor extracting the phasor of high-frequency oscillation signals.

Example Design Procedure for Phasor Estimation

In one example embodiment, a general design method for phasor estimationin different applications is described. The design method can be carriedout offline using a computer, and the designed phasor algorithm can beimplemented in a PMU prototype to estimate the synchrophasor in realtime. FIG. 1 shows an example method 100 for estimating the phasor.

In a first step, the computer can obtain the measurement requirements,e.g., a reporting rate 101, a measurement bandwidth 102, and errorlimitations 103 under static and dynamic conditions. For example, thecomputer can receive limits or upper bounds for total vector error(TVE), amplitude error (AE), phase error (PE) (collectively, 104),frequency error (FE) (105) and rate-of-change-of-frequency error (RFE)(106). More specifically, the computer can receive these error limitsfor the passband range and stopband range, i.e., TVE_(pass)(TVE_(stop)), AE_(pass) (AE_(stop)), PE_(pass) (PE_(stop)), FE_(pass)(FE_(stop)), and RFE_(pass) (RFE_(stop)).

In a second step, the computer can calculate the parameters for acomplex band-pass filter. The filter parameters include the passbandrange 111, passband gain 114, stopband range 112, and stopband gain 115.For each class of PMUs, the passband range and stopband range can beobtained using the reporting rate and the measurement bandwidth. Thepassband range is equal to the measurement bandwidth. Let F_(r) denotethe reporting rate, and thus the stopband range is 0˜(50−F_(r)/2) Hz andover (50+F_(r)/2) Hz. For example, when the reporting rate is 50 Hz, themeasurement bandwidth of M-class PMU is 45 Hz˜55 Hz. Therefore, thepassband range of the complex bandpass filter is 45 Hz˜55 Hz, and thestopband range is 0 Hz-25 Hz and over 75 Hz.

The upper bounds of the passband gain (δ) and the stopband gain (λ) canbe determined using error models 113. In particular, the passband gain(δ) for the phase modulation test is the minimum range obtained usingthe following:

$\begin{matrix}{\delta \leq \frac{{\left( {{PE}_{pass} + k_{p}} \right){J_{0}\left( k_{p} \right)}} - {2\;{J_{1}\left( k_{p} \right)}}}{{\left( {{PE}_{pass} + k_{p}} \right){J_{0}\left( k_{p} \right)}} + {2{J_{1}\left( k_{p} \right)}}}} & (52) \\{\delta \leq {\left( {{TVE}_{pass} - {TVE}_{2}} \right)/\left( {{J_{0}\left( k_{p} \right)} + {2{J_{1}\left( k_{p} \right)}}} \right)}} & (53) \\{\delta \leq \frac{{\left( {{2{\pi\; \cdot {FE}_{pass}}{D\left( f_{m} \right)}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} - {2{J_{1}\left( k_{p} \right)}}}{{\left( {{2\;{\pi \cdot {{FE}_{pass}/{D\left( f_{m} \right)}}}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} + {2{J_{1}\left( k_{p} \right)}}}} & (54)\end{matrix}$where J_(n)(k_(p)) is a first Bessel function of n order, k_(p) is thephase modulation depth, f_(m) is the modulation frequency, TVE₂ dependson f₀±2f_(m), f₀±3f_(m), (f₀ is the fundamental frequency), and D(f) isthe magnitude response of a differentiator used to estimate a frequency.The range of passband gain for other test signals can be determinedusing the same analysis.

Moreover, the stopband gain (λ) for the negative fundamental componentis the minimum range obtained using the following:

$\begin{matrix}{{{H\left( f_{i} \right)}} \leq {\frac{1}{k_{i}}\frac{\tan\;{PE}_{stop}}{\sqrt{1 + {\tan^{2}{PE}_{stop}}}}}} & (55) \\{{{H\left( f_{i} \right)}} \leq {{TVE}_{stop}/k_{i}}} & (56) \\{{{H\left( f_{i} \right)}} \leq {\frac{1}{k_{i}}\frac{2\;{\pi \cdot {{FE}_{stop}/{D\left( {f_{i} - f_{0}} \right)}}}}{1 + {2\;{\pi \cdot {{FE}_{stop}/{D\left( {f_{i} - f_{0}} \right)}}}}}}} & (57)\end{matrix}$Where H(f) is the frequency response of the designed complex bandpassfilter, f_(i) is the frequency of one interference signal, k_(i) is theratio of the magnitude of the interference signal to the fundamentalmagnitude and D(f) is the magnitude response of a differentiator used toestimate a frequency. When the harmonic or inter-harmonic is in thepower signal, the stopband gain can be determined by:

$\begin{matrix}{{E(\lambda)} = {{{{E\left( {{X_{1}^{+}(t)},{z_{1}(t)}} \right)} + {E\left( {{X_{2}^{+}(t)},{z_{2}(t)}} \right)} - {E\left( {{X_{3}^{+}(t)},{z_{3}(t)}} \right)}}} \leq {{{E\left( {{X_{1}^{+}(t)},{z_{1}(t)}} \right)}} + {{E\left( {{X_{2}^{+}(t)},{z_{2}(t)}} \right)}} + {{E\left( {{X_{3}^{+}(t)},{z_{3}(t)}} \right)}}} \leq E_{stop}}} & (58)\end{matrix}$where E(X⁺(t), z(t)) is denoted as the estimation error between X⁺(t)and z(t), and

$\begin{matrix}\begin{matrix}{{z(t)} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( {- f_{0}} \right)}}X_{m}e^{{- j}\; 2\;\pi\; f_{0}t}} +}} \\{{{{H\left( f_{i} \right)}}k_{i}X_{m}e^{j\; 2\;\pi\; f_{i}t}} + {{{H\left( {- f_{i}} \right)}}k_{i}X_{m}e^{{- j}\; 2\;\pi\; f_{i}t}}} \\{= {\left\lbrack {{X_{m}e^{j\; 2\pi\; f_{0}t}} + {{{H\left( {- f_{0}} \right)}}X_{m}e^{{- j}\; 2\;\pi\; f_{0}t}}} \right\rbrack +}} \\{\left\lbrack {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( f_{i} \right)}}k_{i}X_{m}e^{j\; 2\;\pi\; f_{i}t}}} \right\rbrack -} \\{\left\lbrack {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} - {{{H\left( {- f_{i}} \right)}}k_{i}X_{m}e^{{- j}\; 2\;\pi\; f_{i}t}}} \right\rbrack} \\{= {{z_{1}(t)} + {z_{2}(t)} - {z_{3}(t)}}}\end{matrix} & (59) \\\begin{matrix}{{X^{+}(t)} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {X_{m}e^{j\; 2\pi\; f_{0}t}} - {X_{m}e^{j\; 2\;\pi\; f_{0}t}}}}} \\{= {{X_{1}^{+}(t)} + {X_{2}^{+}(t)} - {X_{3}^{+}(t)}}}\end{matrix} & (60)\end{matrix}$where X_(m) is the magnitude of the fundamental signal.

In a third step, the computer can design a complex band-pass filter. Inparticular, using the passband range, passband gain, stopband range, andstopband gain a complex band-pass filter can be designed using the WLSalgorithm or iterative reweighting WLS (IRWLS) algorithm. The WLS orIRWLS algorithm is an optimal design method and can customize the gainat different frequency bands. In addition, these two algorithms havebetter filtering characteristics than other design methods (such aswindow function method and frequency sampling method) under the samefilter order.

In a fourth step, the computer can check the response time and reportinglatency by simulation tests. The response time is the time to transitionbetween two steady-state measurements before and after a step change isapplied to the input. It shall be determined as the difference betweenthe time that the measurement leaves a specified accuracy limit and thetime it reenters and stays within that limit when a step change isapplied to the PMU input. The reporting latency is defined as themaximum time interval between the data report time as indicated by thedata time stamp, and the time when the data becomes available at the PMUoutput. In particular, after designing a complex band-pass filter, onemay check whether the filter's time response and latency depending ondata window meet the application requirements. If the filter's timeresponse and latency depending on data window meet the applicationrequirements, the PMU prototype may use the complex band-pass filter toestimate the phasor. The phasor estimation filter h(k) can be used todevelop the phasor algorithm, and then the synchrophasor can beestimated. In particular, when the band-pass filter coefficients areh(k) (0≤k≤2N, where 2N is the filter order), a positive fundamentalcomponent can be obtained by:

$\begin{matrix}{{z(k)} = {\sum\limits_{i = 0}^{2\; N}{{h(i)}{y\left( {k - N + i} \right)}}}} & (61)\end{matrix}$where y(k) is a discrete power signal, and z(k) is a measured positivefundamental component.

Additionally, the synchrophasor can be obtained by:{dot over (X)}(k)=z(k)e ^(−j2πf) ^(n) ^(t) ^(k)   (62)where {dot over (X)}(k) is the discrete synchrophasor, f_(n) is thenominal frequency, and t_(k) is the report time.

If the filter's time response or latency depending on data window do notmeet the application requirements, the complex band-pass filter may beredesigned after adjusting related requirements. For example, the errorlimitations of the passband and stopband (E_(pass) and E_(stop)) can beadjusted. Then, based on the new error limitations, the complex bandpassfilter is redesigned according to the above steps.

Using the phasor algorithm, the synchrophasor can be estimated invarious classes of the PMU. These classes can include the P-class PMUfor protection and control applications, M-class PMU for widermeasurement bandwidth, and S-class PMU for monitoring thesub-synchronous oscillation.

Hardware Implementation of a PMU Prototype

A PMU may be tested using a high-precision signal generator todemonstrate the practical application of the disclosed methods. FIG. 6shows an example PMU prototype according to an example embodiment. ThePMU prototype 610 can consist of a time-synchronization module 1511, avoltage acquisition module 612, a current acquisition module 613, and acontroller module 614. In one example embodiment, NI 9467, NI 9244, NI9246, and NI cRIO 9039 from the National Instruments can be selected asPMU prototype hardware.

The NI 9467 is a synchronous timing board that can receive GPS signals;it sends one pulse per second (PPS) to generate a 10 kHz sampling clock.The NI 9244 is a voltage acquisition board with a ±400 V input range and4 channels; the NI 9246 is a current acquisition board with a ±20 Ainput range and 3 channels. The NI 9244 and 9246 are 24-bit A/Dconverters and have a maximum sampling frequency of 50 kHz. The NI cRIO9039 is a real-time embedded controller with 2 GB RAM and a 1.91 GHzprocessor. It has eight extended slots to install boards and a built-infield-programmable gate array (FPGA) terminal to process the sampleddata rapidly. The different classes of the phasor estimation methods canbe programmed into the controller to measure the phasor, frequency, andROCOF.

A high-accuracy signal generator (e.g., Omicron CMC256-plus) can be usedas the reference generator to verify the estimation accuracy of the PMUprototype. A computer can control the signal generator to send out testsignals and produce reference measurements according to the mathematicalmodel of the test signal. The mathematical models can be defined in thePMU standards, and the reference phasor, frequency, and ROCOF can bederived from them. Then, the PMU prototype samples the test signals andestimates the measurements. Finally, the estimation accuracy can beobtained by comparing the reference data and the measurements. Thesignal generator and PMU prototype can be synchronized with GPS.

Technical Implementation of a Controller

FIG. 7 illustrates exemplary hardware components of a controller, whichcan be a computer system. A computer system 700, or other computersystems similarly configured, may include and execute one or moresubsystem components to perform functions described herein, includingthe steps of various flow processes described above. Likewise, a mobiledevice, a cell phone, a smartphone, a laptop, a desktop, a notebook, atablet, a wearable device, a server, etc., which includes some of thesame components of the computer system 700, may run an application (orsoftware) and perform the steps and functionalities described above.Computer system 700 may connect to a network 714, e.g., Internet, orother network, to receive inquiries, obtain data, and transmitinformation and incentives as described above.

The computer system 700 typically includes a memory 702, a secondarystorage device 704, and a processor 706. The computer system 700 mayalso include a plurality of processors 706 and be configured as aplurality of, e.g., bladed servers, or other known serverconfigurations. The computer system 700 may also include a networkconnection device 708, a display device 710, and an input device 712.

The memory 702 may include RAM or similar types of memory, and it maystore one or more applications for execution by processor 706. Secondarystorage device 704 may include a hard disk drive, floppy disk drive,CD-ROM drive, or other types of non-volatile data storage. Processor 706executes the application(s), such as those described herein, which arestored in memory 702 or secondary storage 704, or received from theInternet or other network 714. The processing by processor 706 may beimplemented in software, such as software modules, for execution bycomputers or other machines. These applications preferably includeinstructions executable to perform the system and subsystem componentfunctions and methods described above and illustrated in the FIGS.herein. The applications preferably provide graphical user interfaces(GUIs) through which users may view and interact with subsystemcomponents.

The computer system 700 may store one or more database structures in thesecondary storage 704, for example, for storing and maintaining theinformation necessary to perform the above-described functions.Alternatively, such information may be in storage devices separate fromthese components.

Also, as noted, processor 706 may execute one or more softwareapplications to provide the functions described in this specification,specifically to execute and perform the steps and functions in theprocess flows described above. Such processes may be implemented insoftware, such as software modules, for execution by computers or othermachines. The GUIs may be formatted, for example, as web pages inHyperText Markup Language (HTML), Extensible Markup Language (XML) or inany other suitable form for presentation on a display device dependingupon applications used by users to interact with the computer system700.

The input device 712 may include any device for entering informationinto the computer system 700, such as a touch-screen, keyboard, mouse,cursor-control device, microphone, digital camera, video recorder orcamcorder. The input and output device 712 may be used to enterinformation into GUIs during performance of the methods described above.The display device 710 may include any type of device for presentingvisual information such as, for example, a computer monitor orflat-screen display (or mobile device screen). The display device 710may display the GUIs and/or output from sub-system components (orsoftware).

Examples of the computer system 700 include dedicated server computers,such as bladed servers, personal computers, laptop computers, notebookcomputers, palm top computers, network computers, mobile devices, or anyprocessor-controlled device capable of executing a web browser or othertype of application for interacting with the system.

Although only one computer system 700 is shown in detail, system 700 mayuse multiple computer systems or servers as necessary or desired tosupport the users and may also use back-up or redundant servers toprevent network downtime in the event of a failure of a particularserver. In addition, although computer system 700 is depicted withvarious components, one skilled in the art will appreciate that thesystem can contain additional or different components. In addition,although aspects of an implementation consistent with the above aredescribed as being stored in a memory, one skilled in the art willappreciate that these aspects can also be stored on or read from othertypes of computer program products or computer-readable media, such assecondary storage devices, including hard disks, floppy disks, orCD-ROM; or other forms of RAM or ROM. The computer-readable media mayinclude instructions for controlling the computer system 700, to performa particular method, such as methods described above.

The present disclosure is not to be limited in terms of the particularembodiments described in this application, which are intended asillustrations of various aspects. Many modifications and variations canbe made without departing from its spirit and scope, as may be apparent.Functionally equivalent methods and apparatuses within the scope of thedisclosure, in addition to those enumerated herein, may be apparent fromthe foregoing representative descriptions. Such modifications andvariations are intended to fall within the scope of the appendedrepresentative claims. The present disclosure is to be limited only bythe terms of the appended representative claims, along with the fullscope of equivalents to which such representative claims are entitled.It is also to be understood that the terminology used herein is for thepurpose of describing particular embodiments only, and is not intendedto be limiting.

What is claimed is:
 1. A design method for phasor estimation indifferent classes of phasor measurement units (PMUs) using a computerincluding a computer processor and a computer memory and a PMU prototypeincluding a PMU processor and a PMU memory, the method comprising:receiving, at the computer processor, a reporting rate, a measurementbandwidth, and error limitations including TVE_(pass) (TVE_(stop)),AE_(pass) (AE_(stop)), PE_(pass) (PE_(stop)), FE_(pass) (FE_(stop)), andRFE_(pass) (RFE_(stop)); calculating, using the computer processor,parameters for a complex band-pass filter, wherein: the parametersinclude a passband range, a passband gain (δ) fora phase modulationsignal, a stopband range, and a stopband gain (λ) for a negativefundamental component; the passband range and the stopband range arecalculated using the reporting rate and the measurement bandwidth; thepassband gain (δ) for a phase modulation signal is a minimum rangeobtained using the following: $\begin{matrix}{\delta \leq \frac{{\left( {{PE}_{pass} + k_{p}} \right){J_{0}\left( k_{p} \right)}} - {2{J_{1}\left( k_{p} \right)}}}{{\left( {{PE}_{pass} + k_{p}} \right){J_{0}\left( k_{p} \right)}} + {2{J_{1}\left( k_{p} \right)}}}} & (1) \\{\delta \leq {\left( {{TVE}_{pass} - {TVE}_{2}} \right)/\left( {{J_{0}\left( k_{p} \right)} + {2{J_{1}\left( k_{p} \right)}}} \right)}} & (2) \\{\delta \leq \frac{{\left( {{2\;{\pi\; \cdot {{FE}_{pass}/{D\left( f_{m} \right)}}}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} - {2{J_{1}\left( k_{p} \right)}}}{{\left( {{2\;{\pi \cdot {{FE}_{pass}/{D\left( f_{m} \right)}}}} + k_{p}} \right) \cdot {J_{0}\left( k_{p} \right)}} + {2{J_{1}\left( k_{p} \right)}}}} & (3)\end{matrix}$ where J_(n)(k_(p)) is a first Bessel function of n order,f_(m) is a modulation frequency, k_(p) is a phase modulation depth, TVE₂depends on f₀±2f_(m), f₀±3f_(m), f₀ is a fundamental frequency, and D(f)is a magnitude response of a differentiator used to estimate afrequency; and the stopband gain (λ) for a negative fundamentalcomponent is a minimum range obtained using the following:$\begin{matrix}{{{H\left( f_{i} \right)}} \leq {\frac{1}{k_{i}}\frac{\tan\;{PE}_{stop}}{\sqrt{1 + {\tan^{2}{PE}_{stop}}}}}} & (4) \\{{{H\left( f_{i} \right)}} \leq {{TVE}_{stop}/k_{i}}} & (5) \\{{{H\left( f_{i} \right)}} \leq {\frac{1}{k_{i}}\frac{2\;{\pi \cdot {{FE}_{stop}/{D\left( {f_{i} - f_{0}} \right)}}}}{1 + {2{\pi \cdot {{FE}_{stop}/{D\left( {f_{i} - f_{0}} \right)}}}}}}} & (6)\end{matrix}$ where H(f) is a frequency response of the designed complexbandpass filter, f_(i) is a frequency of one interference signal, k_(i)is a ratio of a magnitude of the interference signal to a fundamentalmagnitude and D(f) is the magnitude response of a differentiator used toestimate a frequency; calculating the stopband gain by: $\begin{matrix}{{E(\lambda)} = {{{{E\left( {{X_{1}^{+}(t)},{z_{1}(t)}} \right)} + {E\left( {{X_{2}^{+}(t)},{z_{2}(t)}} \right)} - {E\left( {{X_{3}^{+}(t)},{z_{3}(t)}} \right)}}} \leq {{{E\left( {{X_{1}^{+}(t)},{z_{1}(t)}} \right)}} + {{E\left( {{X_{2}^{+}(t)},{z_{2}(t)}} \right)}} + {{E\left( {{X_{3}^{+}(t)},{z_{3}(t)}} \right)}}} \leq E_{stop}}} & (7)\end{matrix}$ where E(X⁺(t), z(t)) is denoted as the estimation errorbetween X⁺(t) and z(t), and $\begin{matrix}\begin{matrix}{{z(t)} = {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( {- f_{0}} \right)}}X_{m}e^{{{- j}\; 2\;\pi\; f_{0}t}\;}} +}} \\{{{{H\left( f_{i} \right)}}k_{i}X_{m}e^{j\; 2\;\pi\; f_{i}t}} + {{{H\left( {- f_{i}} \right)}}k_{i}X_{m}e^{{- j}\; 2\;\pi\; f_{i}t}}} \\{= {\left\lbrack {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( {- f_{0}} \right)}}X_{m}e^{{- j}\; 2\;\pi\; f_{0}t}}} \right\rbrack +}} \\{\left\lbrack {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {{{H\left( f_{i} \right)}}k_{i}X_{m}e^{j\; 2\;\pi\; f_{i}\; t}}} \right\rbrack -} \\{\left\lbrack {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} - {{{H\left( {- f_{i}} \right)}}k_{i}X_{m}e^{{- j}\; 2\;\pi\; f_{i}t}}} \right\rbrack} \\{= {{z_{1}(t)} + {z_{2}(t)} - {z_{3}(t)}}}\end{matrix} & (8) \\\begin{matrix}{{X^{+}(t)} = {X_{m}e^{j\; 2\;\pi\; f_{0}t}}} \\{= {{X_{m}e^{j\; 2\;\pi\; f_{0}t}} + {X_{m}e^{j\; 2\;\pi\; f_{0}t}} - {X_{m}e^{j\; 2\;\pi\; f_{0}t}}}} \\{= {{X_{1}^{+}(t)} + {X_{2}^{+}(t)} - {X_{3}^{+}(t)}}}\end{matrix} & (9)\end{matrix}$ where X_(m) is the magnitude of the fundamental signal,designing, using the computer processor, a complex band-pass filterusing an iterative reweighting WLS (IRWLS) algorithm; checking, usingthe computer processor, a response time and a reporting latency; whereinthe reporting latency is a maximum time interval between a data reporttime as indicated by a data time stamp, and a time when the data becomesavailable at a PMU output; if the response time and the reportinglatency match a set of predetermined requirement thresholds, estimating,using the PMU processor: a phasor by: $\begin{matrix}{{z(k)} = {\sum\limits_{i = 0}^{2N}{{h(i)}{y\left( {k - N + i} \right)}}}} & (10)\end{matrix}$ where band-pass filter coefficients are h(k) (0≤k≤2N,where 2N is a filter order), y(k) is a discrete power signal, and z(k)is a measured positive fundamental component; a synchrophasor by:{dot over (X)}(k)=z(k)e ^(−j2πf) ^(n) ^(t) ^(k)   (11) where {dot over(X)}(k) is a discrete synchrophasor, f_(n) is a nominal frequency, andt_(k) is a report time; if the response time and the reporting latencydo not match the set of predetermined requirement thresholds,redesigning the complex band-pass filter using updated error limitationswherein redesigning the complex band-pass filter involves adjusting apassband E_(pass) and a stopband E_(stop); and wherein the design methodis implemented in a P-class PMU for protection and control applications,an M-class PMU for wider measurement bandwidth, or an S-class PMU formonitoring a sub-synchronous oscillation.